Isometry/Orthogonal bijective determinant/Exercise

Let ${\displaystyle {}V}$ be a Euclidean vector space. Let

${\displaystyle \varphi \colon V\longrightarrow V}$

denote a linear mapping with the property that the determinant of ${\displaystyle {}\varphi }$ equals ${\displaystyle {}1}$ or ${\displaystyle {}-1}$. Moreover, ${\displaystyle {}\varphi }$ satisfies the property that orthogonal vectors are mapped to orthogonal vectors. Show that ${\displaystyle {}\varphi }$ is an isometry.